Spontaneous Symmetry Breaking

Mexican hat potential V(φ) = −μ²|φ|² + λ|φ|⁴. A symmetric potential whose ground state spontaneously breaks the symmetry — Goldstone mode (circumferential) and Higgs mode (radial).

Parameters

Higgs Mechanism

When μ²>0: vacuum ring at |φ|=√(μ²/2λ)

Goldstone boson: massless, moves along vacuum manifold.

Higgs boson: massive, radial oscillation.

Mass: m_H² = 4μ²